View Proposal #189
If this proposal belongs to you, you are authorized to update it. Use the menu on the right.
| ID | 189 |
|---|---|
| First Name | Christian |
| Last Name | Roettger |
| Institution | Iowa State University |
| Speaker Category | faculty |
| Title of Talk | Pseudo-Random Walks |
| Abstract | In a recent Monthly article, O'Bryant, Reznick and Serbinowska [ORS] have given some fascinating new insights into the behavior of \[ S_{N}(\alpha) := \sum_{n=1}^N (-1)^{[n\alpha]} \] where [x] is the integer part of x. Since the fractional part of n*\alpha for n=1,2,3,\dots behaves 'random-ish', one can make various guesses and apply classical methods like exponential sums to explore these hypotheses. Remarkably, the guesses are often wrong and the classical methods don't seem to work very well. Instead, [ORS] use continued fractions to obtain sharp and explicit upper and lower bounds for |S_{\alpha}(N)| in terms of \log N, and as a by-product get a way of evaluating S_{\alpha}(N) for large N with amazing efficiency.\\ We will explain that last part of their work. Then we will show how to use exponential sums with a twist that gives a lower bound for |S_{\alpha}(N)| - less explicit, but more general than what the methods from [ORS] give you. And if we omit tedious computations (which we will, and which are only long, not hard), the approach is as clear-cut and beautiful as that using exponential sums to the case of the fractional part of n*\alpha. Lit.: K.~O'Bryant, B.~Reznick, M.~Serbinowska: {\em Almost alternating sums}, Monthly vol.~113/8, pp. 673-688. Prerequisites: only complex exponentials e^{it}. |
| Subject area(s) | Number Theory, Dynamical Systems |
| Suitable for undergraduates? | Yes |
| Day Preference | |
| Computer Needed? | |
| Bringing a laptop? | Y |
| Overhead Needed? | N |
| Software requests | |
| Special Needs | |
| Date Submitted | 4/5/2007 |
| Year | 2007 |